The \(p_3\)  integral can be performed with the help of the identity[1]}
\(\int ^{+\infty }_0 \! dx\frac{\cos {(ax)}}{x+\beta }=-\sin {(a\beta )}\mbox{si}(a\beta )-\cos {(a\beta )}\mbox{ci}(a\beta ) \ ,\) 
